# Questions tagged [legendre-polynomials]

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### What's the convergence condition for the generating function formula of Legendre polynomials?

What is the convergence condition of the next infinite series about the Legendre polynomials $P_n(x)$? $$\frac{1}{\sqrt{1-2xt+t^2}}=\sum_{n=0}^\infty P_n(x)t^n$$ I know it is convergent at least ...
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### Generating function of the product of Legendre polynomials

The generating function of the product of Legendre polynomials for the same $n$ is given by \begin{aligned} \sum_{n=0}^{\infty} z^{n} \mathrm{P}_{n}(\cos \alpha) \mathrm{P}_{n}(\cos \beta)&=\frac{\...
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### Properties of a function $C_\ell(\ell)$ which checks an inequality in ideal case (decreasing assumption) and after estimating impact in general case

Suppose that $X=2 \ell+1, Y=C_{\ell}$, both $X$ and $Y$ are function of $\ell$, $X$ is increasing and $Y$ is assuming to be decreasing. But in reality, my data follow a $C_\ell$ increasing for a small ...
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### Defining Legendre polynomials in terms of a sinusoidal function for $|x| \leq 1$

Would it be possible to define Legendre polynomials in terms of a sinusoidal function for $|x|\leq 1$ in a similar manner to Chebyshev polynomials being defined as $T_n(x) = \cos(n \cos^{-1}(x))$? ...
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### Integral with Legendre polynomial

Let $n$ be an integer $\geq 2$ and let $p_n(x)=\displaystyle \frac{1}{n!}(x^n(1-x)^n)^{(n)}$ be a Legendre polynomial. Let $k$ be a positive integer. I would like know if there is a closed formula (...
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• 563
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According to the so-called formula of Laplace (see, e.g., [1, Theorem 8.21.2]), Legendre polynomials admit the following asymptotic expansion: $$P_n(\cos\theta) = \sqrt{2} (\pi n\sin\theta)^{-\frac12}... • 807 1 vote 0 answers 460 views ### Legendre functions as Hypergeometric functions Is there some approximation or regularization that goes in tacitly in the following equality: Q_{\lambda }(z) = \frac{\sqrt{\pi } \Gamma (\lambda +1)}{2^{\lambda +1} \Gamma \left(\lambda +\frac{3}{2}... • 173 2 votes 0 answers 43 views ### Legendre expansion of r(x) = f(x)/g(x) using a finite number of samples from f(x) and g(x) I have two finite sets of events \{x_1, ..., x_N\} and \{y_1, ..., y_N\} that are sampled from the PDFs f(x) and g(x), respectively, where x \in [-1,+1]. I want to estimate the Legendre ... • 21 1 vote 0 answers 175 views ### Expansion of prolate spheroidal harmonics For two coordinate frames O' and O'' both offset along the z-axis by \pm R respectively, with corresponding offset spherical coordinates r', \theta', r'' and \theta'', and with prolate ... • 563 6 votes 2 answers 936 views ### Recurrence of Legendre polynomial roots/ quadrature points Consider Legendre polynomials p_n (x) on [-1,1]. For each n \in \mathbb{N} we denote the zeros of p_n (x) by \left( x_j ^{(n)} \right) _{j=1} ^n. We know that these roots are distinct, and ... • 3,240 0 votes 1 answer 1k views ### Integrals involving associated legendre polynomials Do the following integrals have a closed-form solution for any integer value of m,l,k and n? \int^{\pi}_{0} P^{m}_{l}\left(\cos\theta\right)P^{n}_{k}\left(\cos\theta\right)\cot\theta d\theta \... 7 votes 2 answers 5k views ### Legendre Polynomial Integral How can I evaluate$$ \int_{-1}^1 P_n(x)P_l(x)x^k dx $$when k is even? Or what might be a source where I could find integrals like this? • 563 6 votes 2 answers 642 views ### Symmetric matrix formula for Gauss-Legendre quadrature While searching the web, I came across the following algorithm for the Gauss-Legendre quadrature. I wasn't able to find a reference or a proof of my own as to why it works. I'll present it, and the ... • 3,240 4 votes 1 answer 685 views ### Reference for the exponential decay of Legendre coefficients In Short: I look for a reference to the proof that the spectral coefficients in the Legendre (or Jacobi) expansion are of exponential decay rate. Longer: If p_n is the n-th Legendre polynomial, ... • 3,240 1 vote 0 answers 445 views ### Legendre equation with homogeneous boundary condition I'm looking for solutions to the Legendre differential equation$$ \frac{d}{dx}\left((1-x^2)\frac{d}{dx}f(x)\right)+(\alpha-1)\alpha f(x)=0 $$with boundary conditions f'(0)=0 and f(1)=0, but ... • 310 16 votes 4 answers 1k views ### Maximum of the Vandermonde determinant / minimum of the logarithmic energy The problem is to find the asymptotics (as n\to\infty) of the maximum (say M_n) of the Vandermonde determinant$$V_n:=\prod_{0\le i<j\le n-1}(a_j-a_i) $$over all a_0,\dots,a_{n-1} such ... • 90.3k 14 votes 2 answers 1k views ### Computing Gauss Legendre quadrature for large N I've been scanning across the web, and haven't found a good method to compute the Gauss Legendre abscissas and weights \{ x_j, w^j \} _{j=1}^N for large N\in\mathbb{N}. My question is how to do it,... • 3,240 0 votes 1 answer 130 views ### Equality cannot hold unless x \in \{-1,1\} and/or Wronskian is not zero [closed] By playing around with assoc. Legendre polynomials, I arrived at$$((l+1)+m) (P_l^m(x))^2+((l+1)-m)(P_{l+1}^m(x))^2 = 2(l+1)x P_l^m(x)P_{l+1}^m(x).$$Now, I want to show that we don't have equality ... 0 votes 1 answer 403 views ### Integral Transform with associated Legendre Function of second kind as kernel In my research the following equation appeared:$$\frac{1}{4\pi}\int_{0}^{1}\frac{t^{s-1}(1-t)^{s-1}}{(\rho-t)^s}dt=\int_0^{\infty} f(a) Q^{i\sqrt{a}}_{s-1}(2\rho-1) da,$$where \rho,s>1, Q^{\... • 15 6 votes 0 answers 282 views ### Legendre polynomials and formal groups Let P_n(x) be Legendre polynomials:$$\frac{1}{\sqrt{1-2tx+t^2}}=\sum\limits_{n=0}^{\infty}P_n(x)t^n.$$Usual arguments from the theory of formal groups allow to prove that for any n$$P_n(x)=Q_n(...
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Let $f$ be an $L^1$-function supported in $[-b, b]$, where $0 < b \le b_0 < 1$. Let $Q_n$ be the normalised Legendre polynomials and let $a_n = \int f(x) Q_n(x) dx$ be the Fourier-Legendre ...